#10963: More functorial constructions
-------------------------------------+-------------------------------------
Reporter: nthiery | Owner: stumpc5
Type: enhancement | Status: needs_info
Priority: major | Milestone: sage-6.2
Component: categories | Resolution:
Keywords: days54 | Merged in:
Authors: Nicolas M. Thiéry | Reviewers: Simon King, Frédéric
Report Upstream: N/A | Chapoton
Branch: | Work issues: merge with #15801
public/ticket/10963-doc- | once things stabilize
distributive | Commit:
Dependencies: #11224, #8327, | d6a0e608f9b4b7023dd742a57e62486833b6fa97
#10193, #12895, #14516, #14722, | Stopgaps:
#13589, #14471, #15069, #15094, |
#11688, #13394, #15150, #15506, |
#15757, #15759 |
-------------------------------------+-------------------------------------
Comment (by nthiery):
Replying to [comment:587 nbruin]:
> Doesn't that mean you construct a free F-module that happens to have
some extra structure thanks to S?
Yes, to be precise, the free F-module FS whose basis is indexed by S
(the linear span of S for short).
> > Same thing for additive magmas/monoids/groups.
>
> Is that the same thing? If S is an additive group and the addition
induces the additive relations on the F-module generators,
In this construction, the addition on S induces the *multiplication*
on FS. E.g., if S is an additive group, FS is the usual group algebra
of S.
> > In other words, in most practical use cases, you indeed get an
> > algebra; actually "the algebra"; hence the name.
>
> No, only when you do this with a monoid. (or a magma if you like your
algebras non-associative)
Or even a partial magma :-) Those are not yet implement but we already
had the need for them!
And of course, by the above, also partial additive magmas.
Those two cases are the only ones where something non trivial is
implemented. So at this point the statement "in most practical use
case" definitely holds :-)
But upcoming use cases include:
- "combinatorial" G-module structure on the linear span of a finite
set S endowed with the action of a group G (or semigroup, ...).
- Module structure on the linear span of a crystal S
- ...
> > Still we need a uniform name,
>
> I don't think you do. The construction is nice and functorial over
multiplicative monoids (magmas if you want to treat associativity as an
axiom).
>
> Outside monoids there is a construction that produces the same setwise
result (the free F-module), but it's a functor into a different category.
>
> F-algebras do not form a full subcategory of F-modules, so treating the
two functors as separate makes a lot of sense.
Yes, like CartesianProducts, TensorProducts, and the like, it's indeed
not a functor, but a functorial construction (i.e. a collection of
functors). By default, the functor associated to the category of the
object is applied, but you can specify which functor you want to apply
by specifying the category.
> (it doesn't have much to do with this ticket, though).
Yup.
Cheers,
Nicolas
--
Ticket URL: <http://trac.sagemath.org/ticket/10963#comment:593>
Sage <http://www.sagemath.org>
Sage: Creating a Viable Open Source Alternative to Magma, Maple, Mathematica,
and MATLAB
--
You received this message because you are subscribed to the Google Groups
"sage-trac" group.
To unsubscribe from this group and stop receiving emails from it, send an email
to [email protected].
To post to this group, send email to [email protected].
Visit this group at http://groups.google.com/group/sage-trac.
For more options, visit https://groups.google.com/d/optout.
